A mechanical resonator in a related art will be discussed with reference to FIG. 18. FIG. 18 is a drawing to schematically show the configuration of a “mechanical vibration filter using flexural vibration” introduced in non-patent document 1.
The filter is formed by forming a pattern on a silicon substrate using a thin film process, and is made up of an input line 104, an output line 105, clamped-clamped beams 101 and 102 placed with a gap of 1 micron or less relative to the lines, and a coupling beams 103 for coupling the two beams. A signal input from the input line 104 is capacitively coupled with the beam 101, generating an electrostatic force on the beam 101. Only when the signal frequency matches a frequency close to the resonance frequency of an elastic structure made up of the beams 101 and 102 and the coupling beam 103, mechanical vibration is excited and is detected further as change in the electrostatic capacity between the output line 105 and the beam 102, whereby filtering output of the input signal is taken out.
For a clamped-clamped beam rectangular in cross section, letting elastic modulus be E, density be ρ, thickness be h, and length be L, resonance frequency f of flexural vibration becomes the following expression:
                    f        =                  1.03          ⁢                                          ⁢                      h                          L              2                                ⁢                                    E              ρ                                                          [                  Expression          ⁢                                          ⁢          1                ]            
Letting the material be polysilicon, E=160 GPa and ρ=2.2×103 kg/m3 and letting dimensions be L=40 μm and h=1.5 μm, f=8.2 MHz, and a filter of about an 8-MHz band can be formed. As compared with a filter made up of passive circuits of a capacitor, a coil, etc., a steep frequency selection characteristic with a high Q value can be provided as mechanical resonance is used.
In the described configuration, however, to form a filter of a higher frequency band, the following restriction exists: It is obvious from (expression 1) that it is desirable that first the material should be changed for increasing E/ρ, but if E is increased, the beam displacement amount lessens if the beam deflecting force is the same, and it becomes difficult to detect beam displacement. Letting the index to represent the ease of bending a beam be the ratio between flexural amount d of the beam center part when static load is imposed on the beam surface of a clamped-clamped beam and beam length L, d/L, d/L is represented by the proportional relation of the following expression:
                              d          L                ∝                                            L              3                                      h              3                                ·                      1            E                                              [                  Expression          ⁢                                          ⁢          2                ]            
Thus, to raise the resonance frequency while holding the value of d/L, at least E cannot be changed and a material with low density ρ needs to be found; a composite material of CFRP (Carbon Fiber Reinforced Plastics), etc., needs to be used as a material with low density and Young's modulus equal to polysilicon. In this case, it becomes difficult to form a minute mechanical vibration filter in a semiconductor process.
Then, as a second method not using such a composite material, the beam dimensions can be changed for increasing h·L−2 in (expression 1). However, increasing the beam thickness h and lessening the beam length L lead to lessening the index of the ease of flexure, d/L in (expression 2), and it becomes difficult to detect beam flexure.
As for (expression 1) and (expression 2), the relation between log(L) and log(h) is shown in FIG. 19. A line 191 is relationship found from (expression 1) and a line 192 is relationship found from (expression 2). In FIG. 19, if L and h in the range (region A) above the line with gradient “2” with the current dimension A point as the origin are selected, f increases; if L and h in the range (region B) below the line with gradient “1” are selected, d/L increases. Therefore, the hatched portion in the figure (region C) is the range of L and h in which the resonance frequency can be raised while the beam flexural amount is also ensured.
It is evident from FIG. 19 that microminiaturization of the dimensions of both of the beam length L and the beam thickness h is a necessary condition for putting a mechanical vibration filter into a high frequency and that miniaturization of L and h on the same scaling, namely, lessening L and h while riding on the line with the gradient 1 is a sufficient condition for the hatched portion in FIG. 19.
Thus, in the mechanical resonator in the related art, the dimensions of the mechanical vibrator are miniaturized, whereby the resonance frequency is put into a high frequency. However, generally as the dimensions are miniaturized, the mechanical Q value of flexural vibration lowers; this is a problem. About this phenomenon, for example, non-patent document 2 shows the result of measuring the relationship among the beam length, the beam thickness, and the Q value of flexural resonance using a monocrystalline silicon cantilever. Non-patent document 2 shows that the Q value lowers as the beam length is shortened and the beam thickness is decreased. Therefore, if the resonator using flexural vibration in the related art is miniaturized and is applied to a filter, the Q value required for providing a desirable frequency selection characteristic may be unable to be obtained.
Then, a torsional resonator using a torsional vibrator as a resonator having an excellent Q value is considered. The torsional resonator excites vibrator by an electrostatic force between an input line 204 and a paddle 202 using vibrator 201 having the paddle 202 in the clamped-clamped beam center and converts change in the electrostatic capacity between an output line 205 and the paddle 202 into an electric signal, for example, as shown in FIG. 21. In the torsional resonator, the excited mode varies depending on the magnitude of each of voltage Vi between the paddle and the input line and voltage Vo between the paddle 202 and the output line 205 and the phase difference. Now, assuming that |Vi|>|Vo| and that participation of the electrostatic force of Vo in beam vibration is extremely small, flexural vibration and torsional vibration are excited in the vibrator.
Thus, in addition to torsional vibration, flexural vibration is also excited in the electrode for exciting torsional vibration in the torsional resonator using the torsional vibrator and if such a torsional resonator is applied to a filter, the filter is provided with an unintentional passage band in addition to a passage band.
Differential—signaling described in patent document 1 is available as a method of selectively exciting a specific vibration mode. This is an excitation method for higher-order flexural vibration having a plurality of antinodes; one of an AC signal v and a −v signal different in phase 180 degrees from the AC signal v is applied to the electrode brought close to the antinode and the other is applied to the electrode brought away from the antinode. To produce the AC signal −v from the AC signal v, it is a common practice to divide v and pass through a 180-degree phase shifter.    Non-patent document 1: Frank D. Bannon III, John R. Clark, and Clark T.-C. Nguyen, “High-QHF Microelectromechanical Filters,” IEEE Journal of Solid-State Circuits, Vol. 35, No. 4, pp. 512-526, April 2000.    Non-patent document 2: K. Y. Yasumura et al., “Quality Factors in Micron- and Submicron-Thick Cantilevers,” IEEE Journal of Microelectromechanical Systems, Vol. 9, No. 1, March 2000.    Patent document 1: JP-A-2002-535865 (p 20-p 21, FIG. 8)